Relation between Electric Field and Potential Gradient

In electrostatics, the electric field (E\mathbf{E}) and electric potential (V) are closely related. The electric field at a point is defined as the negative gradient of the electric potential.

Mathematical Derivation:

The electric field is given by:

E=−dVdr\mathbf{E} = -\frac{dV}{dr}

where:

• E\mathbf{E} = Electric field (V/m)
• dVdV = Change in electric potential (V)
• drdr = Small displacement in the direction of the field (m)

This equation shows that the electric field is the rate of change of potential with distance. The negative sign indicates that the electric field points in the direction of decreasing potential.

Vector Form (Three Dimensions):

In three-dimensional space, the electric field is given by the gradient of potential:

E=−∇V\mathbf{E} = -\nabla V

where ∇V\nabla V (del operator) is:

∇V=∂V∂xi^+∂V∂yj^+∂V∂zk^\nabla V = \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k}

Thus,

E=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\mathbf{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)

This means that the electric field points in the direction where the potential decreases most rapidly.

Key Takeaways:

1. Electric field is the negative gradient of potential (E=−dVdr\mathbf{E} = -\frac{dV}{dr}).
2. Higher potential difference means a stronger electric field.
3. Electric field always points from higher potential to lower potential.
4. In a uniform field, E=VdE = \frac{V}{d}, where dd is the distance between two equipotential points.

This relation is crucial in electrostatics for solving problems related to capacitors, conductors, and electric potential energy.